Properties of Quadrilaterals

Draw one diagonal across any quadrilateral and it splits into two triangles. No angle gets created or lost in the process, so the four angles must total two lots of 180 — that is, 360 degrees. This holds for every quadrilateral, from a square to a completely irregular one. The practical use is the same as with triangles: three angles always give you the fourth, by subtracting them all from 360. Angles of 90, 100 and 85 leave 85 for the last one.

Six shapes to know. A square has four equal sides and four right angles. A rectangle has opposite sides equal and four right angles. A rhombus has four equal sides with opposite angles equal. A parallelogram has opposite sides parallel and equal. A trapezium has exactly one pair of parallel sides. A kite has two pairs of adjacent equal sides. The important idea is that these names overlap rather than being six separate boxes: a square is also a rectangle, also a rhombus, and also a parallelogram, because it satisfies all of their definitions. So if a question asks 'is this a parallelogram?' about a square, the answer is yes.

Parallelograms need only two rules. Opposite angles are equal, and adjacent angles — the co-interior ones — add to 180. Both follow from the parallel sides, since co-interior angles between parallel lines are supplementary, which is exactly the rule you met with transversals. The consequence is efficient: one angle of a parallelogram gives you all four. If one angle is 70, the opposite one is also 70, and the two neighbours are 110 each. Check your four angles total 360, and you've confirmed the lot.

The diagonals are what let you tell these shapes apart, and 'bisect' means cut each other exactly in half. In a parallelogram the diagonals bisect each other. In a rectangle they bisect each other and are also equal in length. In a rhombus they bisect each other and cross at right angles. A square does all three, being both a rectangle and a rhombus. In a kite the diagonals cross at right angles and one bisects the other, but not both. This is how proof questions work: told that a shape's diagonals bisect each other at right angles and are equal, you can conclude it's a square.

Putting it together. Given a parallelogram with adjacent angles 3x and 2x, use the co-interior rule: 3x + 2x = 180, so 5x = 180 and x = 36. The four angles come out as 108, 72, 108 and 72, which total 360 — so the answer checks. The routine is always the same six moves: name the shape, recall the property you need, write the equation, solve it, check the four angles total 360, and give the reason. And as with triangles, the reason earns marks — 'angle sum of a quadrilateral', 'opposite angles of a parallelogram', 'co-interior angles, AB parallel to DC', 'diagonals of a rhombus'.